Estimating joint friction and tracking error of a robotics end effector

ABSTRACT

A computerized method for estimating joint friction in a joint of a robotic wrist of an end effector. Sensor measurements of force or torque in a transmission that mechanically couples a robotic wrist to an actuator, are produced. Joint friction in a joint of the robotic wrist that is driven by the actuator is computed by applying the sensor measurements of force or torque to a closed form mathematical expression that relates transmission force or torque variables to a joint friction variable. A tracking error of the end effector is also computed, using a closed form mathematical expression that relates the joint friction variable to the tracking error. Other aspects are also described and claimed.

This patent application claims the benefit of the earlier filing date ofU.S. provisional application No. 62/858,937 filed Jun. 7, 2019.

FIELD

Embodiments related to robotic systems are disclosed. More particularly,embodiments related to surgical robotic systems and techniques forestimating joint friction and tracking error for the end effector of asurgical robotic tool are disclosed.

BACKGROUND INFORMATION

Endoscopic surgery involves looking into a patient's body and performingsurgery inside the body using endoscopes and other surgical tools. Forexample, laparoscopic surgery can use a laparascope to access and viewan abdominal cavity. Endoscopic surgery can be performed using manualtools and/or a surgical robotic system having robotically-assistedtools.

A surgical robotic system may be remotely operated by a surgeon tocontrol a robotically-assisted tool located at an operating table. Thesurgeon may use a computer console located in the operating room, or itmay be located in a different city, to command a robot to manipulate thesurgical tool mounted on the operating table. The robotically-controlledsurgical tool can be a grasper mounted on a robotic arm. Accordingly,the surgical robotic system may be controlled by the remote surgeon tograsp tissue during a robotic surgery.

Control of the surgical robotic system may require control inputs fromthe surgeon. For example, the surgeon may hold in her hand a user inputdevice, UID, such as a joystick or a computer mouse that she manipulatesto generate the signals for the control commands that drive actuatorsand thereby control motion of the surgical robotic system components,e.g., a robotic arm, and/or a surgical tool including its end effectorthat is attached to the arm.

SUMMARY

There may be a tracking error between a commanded position x1 (generatedin part from a current state or pose of the UID) at an actuator, and acontrolled position (e.g., angle) x2 of an end effector that ismechanically coupled to the actuator through a transmission (so as to bedriven by the actuator.) It is demonstrated below in detail how thetracking error may be mathematically estimated (e.g., computed by aprogrammed processor), based on values of i) mechanical compliance (orequivalently stiffness or elasticity) of the transmission and ii) jointfriction that may resist the commanded motion of the end effector.

Another aspect of the disclosure here is a method for estimating (e.g.,computing) joint friction, or friction forces, in the joints of arobotic wrist, based on measurements of force through the transmissionthat couples the robotic wrist to its actuator (e.g., cable forcemeasurements, representing tension in the cable.) The estimated jointfriction may then be used in estimating the tracking error of the endeffector.

In one aspect, a problem to be solved is how to determine (e.g.,estimate) the amount of joint friction, and then how to use thatestimated joint friction together with for example a known (e.g.,assumed) tool compliance or elasticity to estimate the tracking error ofa controlled end effector. Solutions to such a problem help the designerof a surgical robotic system to understand and reduce as needed thetracking error. This may be done by the designer selecting suitablevalues for the transmission stiffness (e.g., greater cable stiffness),or seeking ways to reduce friction torques in the robotic wrist (e.g.,by lubrication or by using different materials.)

The above summary does not include an exhaustive list of all aspects ofthe present invention. It is contemplated that the invention includesall systems and methods that can be practiced from all suitablecombinations of the various aspects summarized above, as well as thosedisclosed in the Detailed Description below and particularly pointed outin the claims filed with the application. Such combinations haveparticular advantages not specifically recited in the above summary.

BRIEF DESCRIPTION OF THE DRAWINGS

In various embodiments, description is made with reference to figures.However, certain embodiments may be practiced without one or more ofthese specific details, or in combination with other known methods andconfigurations. The embodiments of the invention are thus illustrated byway of example and not by way of limitation in the figures in which likereferences indicate similar elements. It should be noted that referencesto “an” or “one” embodiment of the invention or aspect of thisdisclosure are not necessarily to the same embodiment or aspect, andthey mean at least one. Also, in the interest of conciseness andreducing the total number of figures, a given figure may be used toillustrate the features of more than one embodiment of the invention,and not all elements in the figure may be required for a givenembodiment.

FIG. 1 is a pictorial view of an example surgical robotic system 100 inan operating arena.

FIG. 2 illustrates the concept of tracking error using an example simplemechanical system having friction and compliance.

FIG. 3A and FIG. 3B depict an example robotic end effector thatcomprises a pair of jaws at a robotic wrist (for pitch degree offreedom), in closed and open states, respectively.

FIG. 4 shows free body diagrams of the pair of jaws and its roboticwrist (for pitch degree of freedom.)

DETAILED DESCRIPTION

In the following description, numerous specific details are set forth,such as specific configurations, dimensions, and processes, in order toprovide a thorough understanding of the embodiments. In other instances,well-known processes and manufacturing techniques have not beendescribed in particular detail in order to not unnecessarily obscure thedescription. Reference throughout this specification to “oneembodiment,” “an embodiment,” or the like, means that a particularfeature, structure, configuration, or characteristic described isincluded in at least one embodiment. Thus, the appearance of the phrase“one embodiment,” “an embodiment,” or the like, in various placesthroughout this specification are not necessarily referring to the sameembodiment. Furthermore, the particular features, structures,configurations, or characteristics may be combined in any suitablemanner in one or more embodiments.

The use of relative terms throughout the description may denote arelative position or direction. For example, “distal” may indicate afirst direction away from a reference point, e.g., away from anoperator. Similarly, “proximal” may indicate a location in a seconddirection opposite to the first direction, e.g., toward the operator.Such terms are provided to establish relative frames of reference,however, and are not intended to limit the use or orientation of adevice to a specific configuration described in the various embodimentsbelow.

FIG. 1 is a pictorial view of an example surgical robotic system 100 inan operating arena. The robotic system 100 includes a user console 120,a control tower 130, and one or more surgical robotic arms 112 at asurgical robotic platform 111, e.g., a table, a bed, etc. The system 100can incorporate any number of devices, tools, or accessories used toperform surgery on a patient 102. For example, the system 100 mayinclude one or more surgical tools 104 used to perform surgery. Asurgical tool 104 may include an end effector that is attached to adistal end of a surgical arm 112, for executing a surgical procedure.

Each surgical tool 104 may be manipulated manually, robotically, orboth, during the surgery. For example, surgical tool 104 may be a toolused to enter, view, or manipulate an internal anatomy of patient 102.In an embodiment, surgical tool 104 is a grasper that can grasp tissueof patient 102. Surgical tool 104 may be controlled manually, by abedside operator 106; or it may be controlled robotically, via actuatedmovement of the surgical robotic arm 112 to which it is attached.Robotic arms 112 are shown as a table-mounted system, but in otherconfigurations the arms 112 may be mounted in a cart, ceiling orsidewall, or in another suitable structural support.

Generally, a remote operator 107, such as a surgeon or other operator,may use the user console 120 to remotely manipulate the arms 112 and/orsurgical tools 104, e.g., by teleoperation. The user console 120 may belocated in the same operating room as the rest of the system 100, asshown in FIG. 1. In other environments however, the user console 120 maybe located in an adjacent or nearby room, or it may be at a remotelocation, e.g., in a different building, city, or country. The userconsole 120 may comprise a seat 122, foot-operated controls 124, one ormore handheld user input devices, UIDs 126, and at least one operatordisplay 128 configured to display, for example, a view of the surgicalsite inside patient 102. In the example user console 120, remoteoperator 107 is sitting in seat 122 and viewing the operator display 128while manipulating a foot-operated control 124 and a handheld UID 126 inorder to remotely control the arms 112 and surgical tools 104 (that aremounted on the distal ends of the arms 112). Foot-operated control(s)124 can be foot pedals, such as seven pedals, that generate motioncontrol signals when actuated. User console 120 may include one or moreadditional input devices, such as a keyboard or a joystick, to receivemanual inputs to control operations of user console 120 or surgicalrobotic system 100.

In some variations, bedside operator 106 may also operate system 100 inan “over the bed” mode, in which bedside operator 106 is now at a sideof patient 102 and is simultaneously manipulating a robotically-driventool (its end effector, attached to arm 112), e.g., with a handheld UID126 held in one hand, and a manual laparoscopic tool. For example, thebedside operator's left hand may be manipulating the handheld UID 126 tocontrol a robotic component, while the bedside operator's right hand maybe manipulating a manual laparoscopic tool. Thus, in these variations,bedside operator 106 may perform both robotic-assisted minimallyinvasive surgery and manual laparoscopic surgery on patient 102.

During an example procedure (surgery), patient 102 is prepped and drapedin a sterile fashion, and administered anesthesia. Initial access to thepatient anatomy can be achieved using known techniques, such as byforming an incision in the skin. A trocar and/or other surgical tool canbe inserted into the incision through the optical entry in the patient.The trocar can then be positioned at the surgical site (inside the bodyof the patient.) Initial access to the surgical site may be performedmanually while the arms of the robotic system 100 are in a stowedconfiguration or withdrawn configuration (to facilitate access to thesurgical site) or in an operator-defined parking pose. Once initialaccess is completed, initial positioning or preparation of the roboticsystem including docking the arms 112 to their trocars, respectively,and attaching tools (having end effectors) to the arms 112,respectively, may be performed. Next, the surgery proceeds with theremote operator 107 at the user console 120 utilizing the foot-operatedcontrols 124 and the UIDs 126 to manipulate the various end effectorsand perhaps an imaging system, to perform the surgery. Manual assistancemay also be provided at the procedure bed or table, by sterile-gownedbedside personnel, e.g., bedside operator 106 who may perform tasks suchas retracting tissues, performing manual repositioning, and toolexchange upon one or more of the robotic arms 112. Non-sterile personnelmay also be present to assist remote operator 107 at the user console120. When the procedure or surgery is completed, the system 100 and/oruser console 120 may be configured or set in a state to facilitatepost-operative procedures such as cleaning or sterilization andhealthcare record entry or printout via user console 120.

In one embodiment, remote operator 107 holds and moves UID 126 toprovide an input command to move a robot arm actuator 114 in roboticsystem 100. UID 126 may be communicatively coupled to the rest ofrobotic system 100, e.g., via a console computer system 110. UID 126 cangenerate spatial state signals corresponding to movement of UID 126,e.g., position and orientation of the handheld housing of the UID, andthe spatial state signals may be input signals to control a motion ofthe robot arm actuator 114. Robotic system 100 may produce controlsignals as a function of the spatial state signals, to controlproportional motion of actuator 114. In one embodiment, a consoleprocessor of console computer system 110 receives the spatial statesignals and generates the corresponding control signals. Based on thesecontrol signals, which control how the actuator 114 is energized to movea segment or link of arm 112, the movement of a corresponding surgicaltool including an end effector that is attached to the arm may mimic themovement of UID 126. Similarly, interaction between remote operator 107and UID 126 can generate, for example, a grip control signal that drivesan actuator in the arm 112 which in turn causes a jaw of a grasper ofthe surgical tool (that is attached to the arm 112) to close and gripthe tissue of patient 102.

The sensed motion of UID 126 may alternatively be provided to controlother aspects of surgical robotic system 100. For example, gesturesdetected by a finger clutch may generate a clutch signal to pause themotion of actuator 114 and the corresponding surgical tool 104. Forexample, when an operator touches the finger clutch of UID 126 with afinger, the finger clutch may generate a clutch signal, and the clutchsignal may be an input signal to pause the motion of actuator 114.Similarly, one or more capacitive sensing pads may be located on UID126, and the operator may touch the capacitive sensing pads to control acamera view of an endoscope, a cursor on a display of user console 120,etc., while performing a diagnostic, surgical, laparoscopic, orminimally invasive surgical procedure, or another robotic procedure.

Surgical robotic system 100 may include several UIDs 126 whererespective control signals are generated for each UID that control theactuators and the surgical tool (end effector) of a respective arm 112.For example, remote operator 107 may move a first UID 126 to control themotion of actuator 114 that is in a left robotic arm, where the actuatorresponds by moving linkages, gears, etc., in that arm 112. Similarly,movement of a second UID 126 by remote operator 107 controls the motionof another actuator 114, which in turn moves other linkages, gears,etc., of the robotic system 100. Robotic system 100 may include a rightarm 112 that is secured to the bed or table to the right side of thepatient, and a left arm 112 that is at the left side of the patient. Anactuator 114 may include one or more motors, for example housed in thearm including within a tool drive housing on the arm 112. A mechanicaltransmission (partly in the arm 112 and in particular in the tool drivehousing, and partly in a housing of the surgical tool) couples theactuator motors to the joints of the end effector. The actuator motorsare controlled so that they drive the rotation of the joints of the endeffector to change an orientation of the grasper (part of the endeffector.) Motion of several actuators 114 in the same arm 112 can becontrolled by the spatial state signals generated from a particular UID126. UIDs 126 can also be used to translate a finger squeeze into acorresponding motion of the jaws of the grasper. For example, a UID 126can generate a grip signal to control motion of an actuator, e.g., alinear actuator, that opens or closes jaws of the grasper (that is at adistal end of the surgical tool) to grip tissue within patient 102.

In some aspects, the communication between platform 111 and user console120 may be through a control tower 130, which may translate operatorcommands that are received from user console 120 (and more particularlyfrom console computer system 110) into robotic control commands that aretransmitted to arms 112 on robotic platform 111. The control tower 130may also transmit status and feedback from platform 111 back to userconsole 120. The communication connections between the robotic platform111, user console 120, and control tower 130 may be via wired and/orwireless links, using any suitable ones of a variety of datacommunication protocols. Any wired connections may be optionally builtinto the floor and/or walls or ceiling of the operating room. Roboticsystem 100 may provide video output to one or more displays, includingdisplays within the operating room as well as remote displays that areaccessible via the Internet or other networks. The video output or feedmay also be encrypted to ensure privacy and all or portions of the videooutput may be saved to a server or electronic healthcare record system.

It will be appreciated that the operating room scene in FIG. 1 isillustrative and may not accurately represent certain medical practices.

Tracking Error

Referring now to FIG. 2, this figure is used to illustrate the conceptof tracking error. It depicts an example, simple system where a force Fis pulling mass m through a cable having stiffness k. The frictionbetween mass and the ground is Ff, and assume that speed is constant (ormass m is negligible compared to F and Ff). As a result, this yieldsF=Ff. The tracking error between commanded position (x1) and controlledposition (x2) is e=x1−x2, which is essentially equal to the cable(spring) elongation. Therefore, the tracking error may be given bye=Ff/k (ignoring external and inertial forces on the mass). This meansthat the following can be implied: error is proportional to friction;error is inversely proportional to stiffness; and if friction is zero(Ff=0), or stiffness is infinite (the cable is rigid), then trackingerror is essentially zero e=0 and in the absence of friction, externalload (applied on m) can play the same role as friction. In such a case,in order to reduce the tracking error one should focus on increasingstiffness, k.

Some Equations

In order to perform control tasks (by a microprocessor configured orprogrammed to do so according to instructions stored in acomputer-readable storage medium such as microelectronic memory) tocontrol movement of an end effector, it is often beneficial to define aconsistent coordinate frame for the joint angles of the end effectorthat are involved in such movement. For example, consider the exampleend effector that includes a robotic wrist coupled to two jaws of agrasper as shown in FIG. 3A and FIG. 3B, and which is described in moredetail in U.S. Pat. No. 10,166,082 which is incorporated by referenceherein. The jaws pivot about an axis 410 where the jaw angle (θ_(jaw))may be defined as the angle between the two jaws 401A and 401B, and theyaw angle (θ_(yaw)) as the angle between the axis 452 and the linebisecting the jaw angle. Therefore:

$\begin{matrix}\left\{ \begin{matrix}{\theta_{pitch}\  = \ \theta_{1}} \\{\theta_{yaw}\  = \ {\frac{1}{2}\left( {\theta_{2} + \theta_{3}} \right)}} \\{\theta_{jaw}\  = \ {\theta_{2} - \theta_{3}}}\end{matrix} \right. & (1)\end{matrix}$

The transformation between angles in FIG. 3B and these newly definedangles are as follows:

$\begin{matrix}{\begin{bmatrix}\theta_{pitch} \\\theta_{yaw} \\\theta_{jaw}\end{bmatrix} = {D \cdot \begin{bmatrix}\theta_{1} \\\theta_{2} \\\theta_{3}\end{bmatrix}}} & \left( {2.a} \right) \\{D = \begin{bmatrix}1 & 0 & 0 \\0 & {1\text{/}2} & {1\text{/}2} \\0 & 1 & {- 1}\end{bmatrix}} & \left( {2.b} \right)\end{matrix}$

We call this newly defined coordinate system, the extrinsic jointangles. Furthermore, the following nomenclature can be established forpulley geometries:

-   -   a)_(r11) is the radius of the outer pulleys 425A and 425C on        which cables 405A and 405C are residing, respectively;    -   b) r₁₂ is the radius of the inner pulleys 425B and 425D on which        cables 405B and 405D are residing, respectively (τ₁₁ may or may        not be equal to r₁₂);    -   c) r₂₁ is the radius of pulley 415A on the side that cable 405A        is residing (with reference to the center of pulley 415A and        axle 412 as shown in FIG. 3A);    -   d) r₂₂ is the radius of pulley 415A on the side that cable 405B        is residing (with reference to the center of pulley 415A and        axle 412 as shown in FIG. 3A);    -   e) r₃₁ is the radius of pulley 415B on the side that cable 405C        is residing; and    -   f) r₃₂ is the radius of pulley 415B on the side that cable 405D        is residing.

While in the above example the design is symmetrical, r₃₁=r₂₁, r₃₂=r₂₂and r₂₁≠r₂₂ (as shown in FIG. 3A), in some other designs it is possibleto have r₃₁=r₂₁=r₃₂=r₂₂, as wells as r₁₁=r₁₂.

The following equation relates cable tensions or forces in thetransmission, Zeta, (ξ_([4×1])) to joint torques, Tau, (τ_([3×1]))

Tau_([3×1]) =B _([3×4])·Zeta_([4×1])  (3)

where matrix (B) has the following form

$\begin{matrix}{B_{\lbrack{3 \times 4}\rbrack} = \begin{bmatrix}{- r_{11}} & {- r_{12}} & r_{11} & r_{12} \\{- r_{21}} & r_{22} & 0 & 0 \\0 & 0 & r_{31} & {- r_{32}}\end{bmatrix}} & (4)\end{matrix}$

-   -   and (ξ₁, ξ2, ξ3, ξ₄) corresponds to cable tensions on cables        405A, 405B, 405C and 405D, respectively.

In Eq. (3), (τ_([3×1])) is the vector of virtual joint torques appliedby the cables, which may cause the joints that are shown FIG. 3a andFIG. 3b to overcome friction and move against the external forces (thatmay be present on the end effector.) Vector (τ_([3×1])) has threecomponents:

τ_([3×1])=[τ₁τ₂τ₃]^(T)  (6)

where (τ₁) is the pitch joint torque, and (τ₂) and (τ₃) are the jointtorques of jaw 401A and jaw 401B, respectively.

The kinematic relationship that relates the ideal cable displacements(assuming no cable elasticity) and jaw angles are as follows:

q _([4×1])=[q ₁ q ₂ q ₃ q ₄]^(T) =B ^(T)·θ_([3×1])  (7)

where (q_([4×1])) is the four-element vector containing the idealdisplacements of cables 405A-405D, and (θ_([3×1])) is the vector ofangles illustrated in FIG. 3B:

θ_([3×1])=[θ₁θ₂θ₃]^(T)  (8)

In the actual case, where the cables are elastic, the actual and idealcable displacements are related as follows:

ξ_([4×1]) =k(x _([4×1]) −B ^(T)·θ_([3×1]))  (9)

Where ξ_([4×1]) is cable tensions, k is the elastic constant of thecables in N/m (assuming all cables are similar) and (x_([4×1]−B)^(T)·θ_([3×1])) is the difference between actual cable displacementsx_([4×1]) and ideal cable displacements B^(T)·θ_([3×1]).

Tracking Error as a Function of Friction and Stiffness

From eq. (7), we get:

θ_([3×1]) =B ^(−T) ·q _([4×)1]  (10)

Using the hooks law for each of the cables, the cable forces are relatedto displacements on the two sides of the cables as follows:

ξ_([4×1]) =k(x _([4×1]) −q _([4×1]))  (11)

where k is the elastic constant of the cables in N/m, a measure ofstiffness (assuming all cables are similar).

Using equations (10) and (11) we obtain:

θ_([3×1]) =B ^(−T)·(x _([4×1]) −k ⁻¹ξ_([4×1]))  (12)

Here θ_([3×1]) is the vector of actual joint angles. On the other hand,since x_([4×1]) is the cable displacement at the proximal end,B^(−T)x_([4×1]) is the joint angle vector that we think we areachieving, using proximal measurements. The difference between theactual joint angles in eq. (12) and these angles (that we think areachieving) are the joint angle errors, as follows:

θ_(e) _([3×1]) =B ^(−T) ·x _([4×1)]−θ_([3×1)]=k ⁻¹ B^(−T)ξ_([4×1])  (13)

From this equation, it is evident that the tracking error is inverselyproportional to stiffness k.

To see how friction affects tracking error, we need to convert the cableforces to joint torques using eq. (3):

$\begin{matrix}{\theta_{e_{\lbrack{3 \times 1}\rbrack}} = {{k^{- 1}B^{- T}\xi_{\lbrack{4 \times 1}\rbrack}} = {{k^{- 1}B^{- T}{B^{- 1} \cdot \tau_{\lbrack{3 \times 1}\rbrack}}} = {{k^{- 1}\left( {BB^{T}} \right)}^{- 1}\tau_{\lbrack{3 \times 1}\rbrack}}}}} & (14)\end{matrix}$

where it is noted that BB^(T) is a nonsingular square matrix of size3×3. Moreover, τ_([3×1]) is the vector of joint torques—see eq. (6).

In order to progress further, we consider the case where there is gripforce but no side load, although that does not affect the more generalapplicability of the case. That case is considered here only for ease ofillustration of the broader concept. A free body diagram of the two jawsand the proximal wrist (for pitch degree of freedom, DOF) is shown inFIG. 4, where ΣF₁ to ΣF₄ and Στ₄ are the composite reaction forces andtorque at the joints, τ_(f1) is friction at the pitch joint, and τ_(f2)and τ_(f3) are the joint frictions for Jaw1 and Jaw2, respectively.Moreover, F_(g) is the grip force between the two jaws, and τ_(i) arethe joint torques. Finally, F_(ext_pitch) and F_(ext_yaw) are theexternal side loads applied at the jaw tips, and are assumed to bedistributed evenly on both jaws.

The equation of motion of the proximal wrist around the pitch axis is asfollows:

τ₁ −τf1−τ_(ext_pitch) =J ₁(θ){umlaut over (θ)}₁ +C ₁(θ,{dot over(θ)})_([1×3]){dot over (θ)}_([3×1]) +G ₁(θ)  (15)

where τ_(ext_pitch) is the external wrench around the pitch axis causedby the external forces on the jaws.

For the jaws, the external load is usually applied through an objectthat is held between the jaws. Therefore, we assume each jaw takes halfof the load. As a result, we get:

τ₂ −LF _(g)τ_(f2)−τ_(ext_yaw)/2=J ₂(θ){umlaut over (θ)}₂+C ₂(θ,{dot over(θ)})_([1×3]){dot over (θ)}_([3×1]+G) ₂ (θ)  (16)

τ₃ +LF _(g)−τ_(f3)−τ_(external)−τ_(ext_yaw)/2=J ₃(θ){umlaut over (θ)}₃+C ₃(θ,{dot over (θ)})_([1×3]){dot over (θ)}_([3×1]) +G ₃(θ)  (17)

In Equations (15-17), the right hand side are composed of the inertial,Coriolis, centrifugal, and gravity terms. However, as the movingcomponents (wrist and jaws) have very small masses and inertias, theseterms are negligible compared to frictional and driving torques. As aresult, the right hand side of equations (15-17) can be neglected.Therefore, we will get:

$\begin{matrix}\left\{ \begin{matrix}{{\tau_{1} - \tau_{f\; 1} - \tau_{e\_ pitch}} = 0} \\{{\tau_{2} - {LF_{g}} - \tau_{f\; 2} - {\tau_{e\_ yaw}\text{/}2}} = 0} \\{{\tau_{3} + {LF_{g}} - \tau_{f\; 3} - {\tau_{e\_ yaw}\text{/}2}} = 0}\end{matrix} \right. & (18)\end{matrix}$

As a result, we find the vector of joint torques to be:

$\begin{matrix}{\tau = \begin{bmatrix}{\tau_{f\; 1} + \tau_{ext\_ pitch}} \\{\tau_{f\; 2} + {LF_{g}} + {\tau_{ext\_ yaw}\text{/}2}} \\{\tau_{f\; 3} - {LF_{g}} + {\tau_{ext\_ yaw}\text{/}2}}\end{bmatrix}} & (19)\end{matrix}$

Substituting this back into eq. (14) we obtain:

$\begin{matrix}{\theta_{e_{\lbrack{3 \times 1}\rbrack}} = {{k^{- 1}\left( {BB^{T}} \right)}^{- 1}\begin{bmatrix}{\tau_{f\; 1} + \tau_{e\_ pitch}} \\{\tau_{f\; 2} + {LF_{g}} + {\tau_{ext\_ yaw}\text{/}2}} \\{\tau_{f\; 3} - {LF_{g}} + {\tau_{ext\_ yaw}\text{/}2}}\end{bmatrix}}} & (20)\end{matrix}$

Using eq. (2), the error angles in extrinsic coordinate system will beas follows:

$\begin{matrix}{\begin{bmatrix}\theta_{e\_ pitch} \\\theta_{e\_ yaw} \\\theta_{e\_ jaw}\end{bmatrix} = {D\theta_{e_{\lbrack{3 \times 1}\rbrack}}}} & (21)\end{matrix}$

Substituting eq. (20) we get:

$\begin{matrix}{\begin{bmatrix}\theta_{e\_ pitch} \\\theta_{e\_ yaw} \\\theta_{e\_ jaw}\end{bmatrix} = {k^{- 1}{{D\left( {BB^{T}} \right)}^{- 1}\begin{bmatrix}{\tau_{f\; 1} + \tau_{e\_ pitch}} \\{\tau_{f\; 2} + {LF_{g}} + {\tau_{ext\_ yaw}\text{/}2}} \\{\tau_{f\; 3} - {LF_{g}} + {\tau_{ext\_ yaw}\text{/}2}}\end{bmatrix}}}} & (22)\end{matrix}$

To separate the effect of friction and grip force on tracking error, theterm in the parenthesis can be decomposed further as follows:

$\begin{matrix}{\begin{bmatrix}\theta_{e\_ pitch} \\\theta_{e\_ yaw} \\\theta_{e\_ jaw}\end{bmatrix} = {{k^{- 1}{{D\left( {BB^{T}} \right)}^{- 1}\begin{bmatrix}\tau_{f\; 1} \\\tau_{f\; 2} \\\tau_{f\; 3}\end{bmatrix}}} + {k^{- 1}{{D\left( {BB^{T}} \right)}^{- 1}\begin{bmatrix}0 \\{+ {LF}_{g}} \\{- {LF}_{g}}\end{bmatrix}}} + {k^{- 1}{{D\left( {BB^{T}} \right)}^{- 1}\begin{bmatrix}\tau_{e\_ pitch} \\{\tau_{ext\_ yaw}\text{/}2} \\{\tau_{ext\_ yaw}\text{/}2}\end{bmatrix}}}}} & (23)\end{matrix}$

The first term on the right hand side of the above equation captures theeffect of friction on extrinsic joint angle errors and shows that thetracking error is linearly proportional to joint friction. The secondand third terms capture the influence of grip and external loads on thetracking error.

Replacing B, and D from eq. (4), and (2.b) respectively, and assumingr₃₁=r₂₁, and r₃₂=r₂₂, we get:

$\begin{matrix}{\begin{bmatrix}\theta_{e\_ pitch} \\\theta_{e\_ yaw} \\\theta_{e\_ jaw}\end{bmatrix} = {{{k^{- 1}\begin{bmatrix}\frac{r_{21}^{2} + r_{22}^{2}}{2\begin{pmatrix}{{r_{11}r_{22}} +} \\{r_{12}r_{21}}\end{pmatrix}^{2}} & \frac{{r_{12}r_{22}} - {r_{11}r_{21}}}{2\begin{pmatrix}{{r_{11}r_{22}} +} \\{r_{12}r_{21}}\end{pmatrix}^{2}} & 0 \\\frac{{r_{12}r_{22}} - {r_{11}r_{21}}}{2\begin{pmatrix}{{r_{11}r_{22}} +} \\{r_{12}r_{21}}\end{pmatrix}^{2}} & \frac{r_{11}^{2} + r_{12}^{2}}{2\begin{pmatrix}{{r_{11}r_{22}} +} \\{r_{12}r_{21}}\end{pmatrix}^{2}} & 0 \\0 & 0 & \frac{1}{r_{21}^{2} + r_{22}^{2}}\end{bmatrix}}\begin{bmatrix}\tau_{f\; 1} \\{\tau_{f\; 2} + \tau_{f\; 3}} \\{\tau_{f\; 2} - \tau_{f\; 3}}\end{bmatrix}} + {{k^{- 1}\begin{bmatrix}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & \frac{2}{r_{21}^{2} + r_{22}^{2}}\end{bmatrix}}\begin{bmatrix}0 \\0 \\{LF_{g}}\end{bmatrix}} + {{k^{- 1}\begin{bmatrix}\frac{r_{21}^{2} + r_{22}^{2}}{2\begin{pmatrix}{{r_{11}r_{22}} +} \\{r_{12}r_{21}}\end{pmatrix}^{2}} & \frac{{r_{12}r_{22}} - {r_{11}r_{21}}}{2\begin{pmatrix}{{r_{11}r_{22}} +} \\{r_{12}r_{21}}\end{pmatrix}^{2}} & 0 \\\frac{{r_{12}r_{22}} - {r_{11}r_{21}}}{2\begin{pmatrix}{{r_{11}r_{22}} +} \\{r_{12}r_{21}}\end{pmatrix}^{2}} & \frac{r_{11}^{2} + r_{12}^{2}}{2\begin{pmatrix}{{r_{11}r_{22}} +} \\{r_{12}r_{21}}\end{pmatrix}^{2}} & 0 \\0 & 0 & 0\end{bmatrix}}\begin{bmatrix}\tau_{{ext\_ pitc}h} \\\tau_{ext\_ yaw} \\0\end{bmatrix}}}} & (24)\end{matrix}$

Several observations can be made from equation (24). As can be seen, thepitch and yaw tracking errors due to friction are only affected byτ_(f1), and τ_(f2)+τ_(f3). On the other hand, the tracking error for jawDOF is only affected by τ_(f2)−τ_(f3). It is also observed that, asexpected, the grip force only affects the jaw DOF. However, we know thatduring grip application, the jaw is closed. So the actual measured jawangle should be zero (or should be the jaw closure angle at contact).What is represented in second term of eq. (20) is the additional amountwe read (as joint status) when the jaws are closed. For joint angleestimation purposes, this value needs to be subtracted from theestimated jaw angle found purely based on cable displacements andinverse kinematics. Moreover, we can consider anapproximation/simplification for the case that a grip force is beingapplied. In this case, since the jaws are moving in the same direction,and same speed, and the forces on the jaw pins are very similar, thejoint friction values will be also similar. As a result, we can assumeτ_(f2)−τ_(f3)≅0, in this special case.

Additionally, from the third term on the right hand side of Eq. (24) weobserve that τ_(ext_pitch), and τ_(ext_yaw) affect the tracking errorvery similarly to τ_(f1), and τ_(f2)+τ_(f3). The essentially play asimilar role in providing a resistive force on the distal end of thedevice. Without knowing the magnitudes of the external loads, andwithout any distal sensors, they are essentially indistinguishable fromjoint frictions.

As seen in Eq. (24), the joint friction values (e.g., Tauf1 which isfriction torque at the pitch joint, and Tauf2 and Tauf3 which are jointfrictions for Jaw1 and Jaw2, respectively) are required for calculatingthe end-effector tracking errors. The friction in the joints areproportional to the normal forces on the joint pins. These forces canchange with the magnitude of grip or external loads on the jaws, therebychanging the amount of friction. In what follows, we seek a relationshipbetween the frictional forces/torques and the cable forces, where thecable forces can be measured using sensors.

As defined in eq. (6), the driving torques can be expressed, using eq.(3), as a function of cable forces:

$\begin{matrix}\left\{ \begin{matrix}{\tau_{1} = {{{- r_{11}}\xi_{1}} - {r_{12}\xi_{2}} + {r_{11}\xi_{3}} + {r_{12}\xi_{4}}}} \\{\tau_{2} = {{{- r_{21}}\xi_{1}} + {r_{22}\xi_{2}}}} \\{\tau_{3} = {{r_{31}\xi_{3}} - {r_{32}\xi_{4}}}}\end{matrix} \right. & (25)\end{matrix}$

Solving equations (25) to eliminate the driving torques, and assumingr21=r31, and r22=r32, we get:

$\begin{matrix}\left\{ \begin{matrix}{\tau_{f\; 1} = {{- {r_{11}\left( {\xi_{1} - \xi_{3}} \right)}} - {r_{12}\left( {\xi_{2} - \xi_{4}} \right)} - \tau_{ext\_ pitch}}} \\{{\tau_{f\; 2} + \tau_{f\; 3}} = {{- {r_{21}\left( {\xi_{1} - \xi_{3}} \right)}} + {r_{22}\left( {\xi_{2} - \xi_{4}} \right)} - \tau_{ext\_ yaw}}} \\{{\tau_{f\; 2} - \tau_{f\; 3}} = {{- {r_{21}\left( {\xi_{1} + \xi_{3}} \right)}} + {r_{22}\left( {\xi_{2} + \xi_{4}} \right)} - {2\; L\; F_{g}}}}\end{matrix} \right. & (26)\end{matrix}$

As discussed above, in the case the jaws are grasping an object andmoving together, we have τ_(f2)−τ_(f3)≅0. Using the third equation ineq. (26), we arrive at an equation that can be used for estimating thegrip force:

$\begin{matrix}{F_{g} = \frac{{- {r_{21}\left( {\xi_{1} + \xi_{3}} \right)}} + {r_{22}\left( {\xi_{2} + \xi_{4}} \right)}}{2L}} & (27)\end{matrix}$

Once cable force measurements are available (by producing sensormeasurements of force in the transmission, that mechanically couples therobotic wrist to its actuator), the joint friction torques can becomputed from equations (26), which is a closed form mathematicalexpression that relates transmission force or torque variables Zeta1,Zeta2, Zeta3, Zeta4 to joint friction variables Tauf1, Tauf2, Tauf3 forjoints of the end effector. The so-computed joint friction values canthen be replaced back into Eq. (24) which is a closed form mathematicalexpression that relates the joint friction variable to the trackingerror, e.g., used to compute the estimated tracking errors

$\quad\begin{matrix}\theta_{e\_ pitch} \\\theta_{e\_ yaw} \\\theta_{e\_ jaw}\end{matrix}$

of the end effector. The designer can then determine suitable values forcable stiffness (k), or may decide to reduce friction torques (e.g., bylubrication, or using different materials), as ways to reduce thetracking error.

While certain aspects have been described above and shown in theaccompanying drawings, it is to be understood that such descriptions aremerely illustrative of and not restrictive on the invention, and thatthe invention is not limited to the specific constructions andarrangements shown and described, since various other modifications mayoccur to those of ordinary skill in the art. For example, while theabove description illustrates the method of estimating joint frictionusing a wristed pair of pivoting jaws, and in particular the jointfriction associated with controlling the jaw angle and pitch and yaw ofa surgical robotic wrist, the method is also applicable to estimatingjoint friction in other types of surgical robotic end effectors, e.g.,needle drivers. The description is thus to be regarded as illustrativeinstead of limiting.

1. A computerized method for estimating joint friction in a joint of arobotic wrist of an end effector, the method comprising: a. producingsensor measurements of force or torque in a transmission thatmechanically couples a robotic wrist to an actuator; and b. computingjoint friction in a joint of the robotic wrist that is driven by theactuator, wherein the joint friction is computed by applying the sensormeasurements of force or torque to a closed form mathematical expressionthat relates transmission force or torque variables to a joint frictionvariable.
 2. The method of claim 1 further comprising a. computing atracking error of the end effector using a closed form mathematicalexpression that relates the joint friction variable to the trackingerror.
 3. The method of claim 2 wherein the sensor measurements arecable force measurements.
 4. The method of claim 3 wherein the cableforce measurements are measurements of force or tension on a pluralityof cables that are part of the transmission that mechanically couplesthe robotic wrist to the actuator.
 5. The method of claim 4 wherein theclosed form mathematical expression relates a plurality of cable tensionvariables to a plurality of joint friction torque variables, wherein theplurality of joint friction torque variables represent friction at aplurality of joints of the end effector.
 6. The method of claim 2wherein the end effector comprises a pair of pivoting jaws having a jawangle between them and being mechanically coupled to the wrist to bedriven by the actuator, and the tracking error refers to the jaw angle.7. The method of claim 6 wherein the joint friction variable representsfriction at the joint of the wrist.
 8. The method of claim 1 wherein thesensor measurements are cable force measurements.
 9. The method of claim8 wherein the cable force measurements are measurements of force ortension on a plurality of cables that are part of the transmission thatmechanically couples the robotic wrist to the actuator.
 10. The methodof claim 9 wherein the closed form mathematical expression relates aplurality of cable tension variables to a plurality of joint frictiontorque variables, wherein the plurality of joint friction torquevariables represent friction at a plurality of joints of the endeffector.
 11. An article of manufacture comprising a computer-readablestorage medium having stored therein instructions that configure aprocessor to estimate joint friction in a joint of a robotic wrist of anend effector that is part of a surgical tool, the method comprising: a.producing sensor measurements of force or torque in a transmission,wherein the transmission mechanically couples a robotic wrist to anactuator so the actuator can drive a joint of the robotic wrist; and b.computing joint friction in the joint of the robotic wrist by applyingthe sensor measurements of force or torque to a closed form mathematicalexpression that relates transmission force or torque variables to ajoint friction variable.
 12. The article of manufacture of claim 11wherein the computer-readable storage medium has stored therein furtherinstructions that configure the processor to compute a tracking error ofthe end effector using a closed form mathematical expression thatrelates the joint friction variable to the tracking error.
 13. Thearticle of manufacture of claim 12 wherein the sensor measurements arecable force measurements.
 14. The article of manufacture of claim 13wherein the cable force measurements are measurements of force ortension on a plurality of cables that are part of the transmission thatmechanically couples the robotic wrist to the actuator.
 15. The articleof manufacture of claim 14 wherein the closed form mathematicalexpression relates a plurality of cable tension variables to a pluralityof joint friction torque variables, wherein the plurality of jointfriction torque variables represent friction at a plurality of joints ofthe end effector.
 16. The article of manufacture of claim 12 wherein theend effector comprises a pair of pivoting jaws having a jaw anglebetween them and being mechanically coupled to the robotic wrist to bedriven by the actuator, and the tracking error refers to the jaw angle.17. The article of manufacture of claim 16 wherein the joint frictionvariable represents friction at the joint of the robotic wrist.
 18. Thearticle of manufacture of claim 11 wherein the sensor measurements arecable force measurements.
 19. The article of manufacture of claim 18wherein the cable force measurements are measurements of force ortension on a plurality of cables that are part of the transmission thatmechanically couples the robotic wrist to the actuator.
 20. The articleof manufacture of claim 19 wherein the closed form mathematicalexpression relates a plurality of cable tension variables to a pluralityof joint friction torque variables, wherein the plurality of jointfriction torque variables represent friction at a plurality of joints ofthe end effector.